Preamble configuring method in the wireless lan system, and a method for a frame synchronization

ABSTRACT

A method of configuring a preamble of a downlink frame for synchronization in data frame transmission of a 60 GHz a wireless local area network system, the method comprising arranging a short preamble having a plurality of repetitive S symbols, and an IS symbol, and arranging a long preamble having a long cyclic prefix (CP) and a plurality of L symbols for frame synchronization and symbol timing by performing auto-correlation according to the length of window of the auto-correlation.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Continuation of U.S. patent application Ser. No.15/686,656 filed on Aug. 25, 2017, which is a is a Continuation of U.S.patent application Ser. No. 13/491,941 filed on Jun. 8, 2012, now U.S.Pat. No. 9,769,002 issued Sep. 19, 2017, which is a Continuation of U.S.patent application Ser. No. 10/584,335 having a 371(c) date of Jun. 23,2006, now U.S. Pat. No. 8,218,427 issued Jul. 10, 2012, which is a U.S.National Stage application of International Application No.PCT/KR2004/003471, filed on Dec. 27, 2004, which claims priority toKorean Patent Application No. 10-2003-98211 filed on Dec. 27, 2003 inthe Korean Intellectual Property Office, the entire disclosures of whichare incorporated herein by reference for all purposes.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention relates to a preamble configuring method and aframe synchronization detection method in a wireless local area networksystem, and more particularly it relates to a preamble configuringmethod and a detection method for data frame synchronization in a 60 GHzwireless local area network system.

Description of the Related Art

During data frame transmission in a conventional wireless local areanetwork (WLAN) system, a preamble signal is transmitted to a receiver atthe beginning of the data frame and the receiver detects the start offrame transmission and performs a symbol timing detection fordemodulation using the preamble signal.

Methods for detecting the symbol timing, an auto-correlation or across-correlation of a received signal may be employed.

The cross-correlation requires lots of calculations in every clockperiod and may cause serious performance degradation due to a carrierfrequency offset, whereas the auto-correlation requires lesscalculations and can be simply implemented.

“Synchronization Technique for HIPERLAN/2” (IEEE 54th VehicularTechnology Conference, vol. 2, p. 762˜766, 2001, by V. Almenar) isrelated to a method for detecting frame synchronization by using thephase and amplitude of auto-correlation in a 5 GHz OFDM (OrthogonalFrequency Division Multiplexing) WLAN system.

However, the above-disclosed transaction may not be available forestimating carrier frequency offset in a 60 GHz WLAN system.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide a method for framesynchronization detection for a 60 GHz wireless local area network(WLAN) system for efficiency of performance and realization.

It is another object of the present invention to provide a method forconfiguring a downlink physical layer preamble and an uplink physicallayer preamble for the 60 GHz WLAN system.

It is another object of the present invention to provide a method forconfiguring preambles and detecting frame synchronization by usingperiodically repeated preambles and auto-correlation to enhanceperformance and simplify complexity of realization.

In one aspect of the present invention, there is provided a method forconfiguring a preamble of a downlink frame for synchronization andchannel estimation in a wireless local area network system. The methodcomprises a) arranging a short preamble at starting points of an uplinkburst and a downlink burst and b) arranging a long preamble used forfine frequency offset estimation and channel estimation in the receiverafter the short preamble. The short preamble is used for time andfrequency synchronization in a receiver. In a), a plurality of S symbolsare repetitively arranged in the starting points of the uplink burst andthe downlink burst, and an IS symbol is arranged after the S symbols. Inb), a long cyclic prefix (CP) is arranged after the short preamble, anda plurality of L symbols are repetitively arranged after the long CP.

In another aspect of the present invention, there is provided a methodfor detecting synchronization of data transmitted per frame in awireless local area network system, the frame comprising a shortpreamble having a plurality of repetitive S symbols, and an IS symbol.In the method, frame synchronization of a short preamble in a form of aperiodically repeated signal is detected according to a characteristicof auto-correlation of the short preamble, and timing is estimated byperforming auto-correlation according to windows having lengths set tohave different periods.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and constitute apart of the specification, illustrate an embodiment of the invention,and, together with the description, serve to explain the principles ofthe invention.

FIG. 1 illustrates a downlink protocol data unit (PDU) train for a 60GHz wireless LAN (WLAN) system according to the embodiment of thepresent invention.

FIG. 2 illustrates a structure of a downlink preamble for the 60 GHzwireless LAN (WLAN) system according to the embodiment of the presentinvention.

FIG. 3 depicts a structure of a short preamble in the time domain forthe 60 GHz WLAN system according to the embodiment of the presentinvention.

FIG. 4 depicts a structure of a long preamble in the time domain for the60 GHz WLAN system according to the embodiment of the present invention.

FIG. 5 depicts parameters of the preamble in a time domain for the 60GHz WLAN system according to the embodiment of the present invention.

FIG. 6 depicts a frame synchronization estimation process in the casethat a cyclic prefix is inserted instead of an IS symbol in the shortpreamble for the 60 GHz WLAN system according to the embodiment of thepresent invention.

FIG. 7 to FIG. 10 depict auto-correlation specified by the length of awindow during frame synchronization performance for the 60 GHz WLANsystem according to the embodiment of the present invention.

FIG. 11 explains a method of symbol timing estimation usingauto-correlation according to the embodiment of the present invention.

FIG. 12 explains a symbol timing detection process, separately includingauto-correlation to reduce the probability of a false alarm (FA).

FIG. 13 depicts a symbol timing detection process includingauto-correlation for confirmation according to the embodiment of thepresent invention.

FIG. 14 explains a frame synchronization process in the case that the ISsymbol is inserted instead of the CP symbol in the short preamble of the60 GHz WLAN system according to the embodiment of the present invention.

FIG. 15 depicts the frame synchronization including a timing estimationprocess to increase timing accuracy according to the embodiment of thepresent invention.

FIG. 16 depicts a simulation result for determining the length of awindow in consideration of the amount of calculation during the framesynchronization performance and fine symbol timing estimation accordingto the embodiment of the present invention

FIG. 17 to FIG. 19 depict results of channel-specified framesynchronization performance when the length of a window is set to be 64samples, and a detection range is set to be 64 (±32) samples accordingto the embodiment of the present invention.

FIG. 20 to FIG. 22 depict results of a frame synchronization algorithmwhen the length of an auto-correlation window is set to be 64 samplesand the detection ranges are respectively set to be 16 samples, 32samples, and 64 samples for the timing estimation process included inthe frame synchronization process.

FIG. 23 shows a table for comparing a frame synchronization algorithmusing an auto-correlation window of a sufficient length (first method)with a frame synchronization including the timing estimation process(second method) when the detection range is equally set to be 16 samplesand a signal-to-noise ratio (SNR) is set to be 5 dB according to thepresent invention.

FIG. 24 shows a table comparing the first method and the second methodwhen the detection range is equally set to be 64 samples and the SNR isset to be 5 dB according to the embodiment of the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following detailed description, only the preferred embodiment ofthe invention has been shown and described, simply by way ofillustration of the best mode contemplated by the inventor(s) ofcarrying out the invention. As will be realized, the invention iscapable of modification in various obvious respects, all withoutdeparting from the invention. Accordingly, the drawings and descriptionare to be regarded as illustrative in nature, and not restrictive.

Referring to the accompanying drawings, methods for configuring apreamble and detecting frame synchronization in a wireless local areanetwork (WLAN) system according to the embodiment of the presentinvention will be described in detail.

In a typical communication system, a transmitter transmits a preamble ora training signal to a receiver to pre-notify data frame transmissionbefore the actual data frame is transmitted. The receiver thereforeperceives that the actual data frame will be transmitted by receivingthe preamble or the training signal.

According to embodiments of the present invention, a downlink physicallayer preamble and an uplink physical layer preamble are implemented,and methods for detecting start of data frame transmission by employingauto-correlation of periodic and continuous preamble signals anddetecting frame synchronization by using the phase and amplitude of theauto-correlation are provided. The frame synchronization detectionincludes timing estimation to enhance timing accuracy.

In the embodiments of the present invention, the frame synchronizationalgorithm is employed because the 60 GHz WLAN system supports theorthogonal frequency division multiplexing (OFDM)/time division duplex(TDD) system. When a synchronization algorithm employed in aHIPERLAN/2-based system is applied to carrier frequency offset of ±5.12of the 60 GHz WLAN system, phase variation becomes 2π×5.12×16(windowdelay)/256(FFT size)=0.64π thus exceeding π/2, and it is accordinglydifficult to distinguish an upper link and a lower link in the 60 GHzWLAN system. For the same reason, a pre-compensation method is usedwithout considering a short preamble of the upper link.

There are two types of physical layer bursts in the 60 GHz WLAN system:a downlink (DL) burst and an uplink (UL) burst. Protocol data unit (PDU)trains input to an upper layer are added with preambles forsynchronization and channel estimation and mapped into the physicallayer (PHY) bursts.

FIG. 1 illustrates a DL PDU train for the 60 GHz WLAN system accordingto an exemplary embodiment of the present invention. The DL PDU trainincludes a preamble 11 and a payload 12. The DL PDU train is mapped tothe DL PHY burst which is generated by adding the preamble 11 to aplurality of baseband OFDM symbols.

FIG. 2 illustrates a structure of a DL preamble for the 60 GHz WLANsystem according to an exemplary embodiment of the present invention.The DL preamble has the length of T_(P), and includes a short preamble21 having the length of T_(sp) and a long preamble 22 having the lengthof T_(LP). The short preamble 21 is used by the receiver for time andfrequency synchronization.

FIG. 3 illustrates a structure of the short preamble in a time domain ofthe 60 GHz WLAN system according to an exemplary embodiment of thepresent invention. The short preamble has S symbols 31 and 32 that arerepeated 16 times within a data symbol period, and an IS symbol 33having the length of a guard interval (hereinafter, referred to ascyclic prefix CP). There is a 180° phase difference between the S symboland the IS symbol.

A frequency domain signal of the short preamble is given as Equation 1.Herein, a time domain signal of the short preamble is formed by addingthe IS symbol to a signal that is an Inverse Fast Fourier Transform(IFFT) processed frequency domain signal.

$\begin{matrix}{{SP}_{k} = \left\{ \begin{matrix}{{\sqrt{\frac{200}{24}} \times \left( {C_{1,{m + 1}}^{4} + {jC}_{8,{m + 1}}^{4}} \right)},} & {{k = {16 \times m}},{0 \leq m < 6}} \\{\sqrt{\frac{200}{24}} \times \left( {C_{1,{m + 1}}^{4} + {jC}_{8,{m + 1}}^{4}} \right)} & \begin{matrix}{{k = {{16 \times \left( {m + 1} \right)} + 4}},} \\{6 \leq m < 11}\end{matrix} \\{0,} & {otherwise}\end{matrix} \right.} & \left\lbrack {{Equation}\mspace{14mu} 1} \right\rbrack\end{matrix}$

Where √{square root over (200/24)} is a normalized power value resultedfrom using 12 sub-carriers among 200 sub-carriers, C_(sm) ⁴ iscalculated by inverting 0 into −1 in a matrix generated by an m-sequencegenerator of a fourth-degree polynomial x⁴+x+1, and s is an initialvalue.

FIG. 4 illustrates a structure of a long preamble in the time domain ofthe 60 GHz WLAN system. The long preamble having the length of T_(LP) isused by the receiver for fine frequency offset estimation and channelestimation. The long preamble includes two L symbols 42 and 43respectively having the length of T_(LP) and a long CP 41. The length ofthe L symbol is twice as long as that of the data symbol period, and thelength of the long CP 41 is twice as long as that of the CP of the ISsymbol 33.

A frequency domain signal of the long preamble is given as Equation 2.Herein, a time domain signal of the long preamble is formed by insertinga signal and the long CP 41 that is twice as long as the CP of the ISsymbol 33, the signal being generated by the IFFT-processing thefrequency domain signal and repeating the IFFT processed frequencydomain signal twice.

$\begin{matrix}{{LP}_{k} = \left\{ {\begin{matrix}{C_{1,{m + 11}}^{8},} & {{{if}\mspace{14mu} k} \neq 100} \\{0,} & {{{if}\mspace{14mu} k} = 100}\end{matrix},{0 \leq k < 200}} \right.} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

Where C_(sm) ⁸ is calculated by inverting 0 into −1 in a matrixgenerated from an m-sequence generator of an eight-degree polynomialx⁸+x⁷+x⁶+x+1, and s denotes an initial value.

FIG. 5 illustrates parameters of the preamble in the time domain of the60 GHz WLAN system. Each wireless terminal (WT) estimates a carrierfrequency by using a DL preamble of a DL sub-frame and maintains the ULcarrier frequency offset within 1% of a difference between thesub-carrier frequencies since the uplink (UL) burst has no periodical ULpreamble. Further, adequate transmission power for the UL is derivedfrom average transmission power of an access point (AP) estimated by theDL preamble. In other words, the UL signal transmitted to the AP doesnot require carrier frequency restoration and automatic gain control(AGC) processes, and the estimated carrier frequency offset is small,such that the CP is used to estimate symbol timing.

The frame synchronization is used to find the start point of each frameand estimate the frame timing for fine symbol timing estimation. Framesynchronization performance is defined by referring to the followingcriteria: timing accuracy, false alarm (FA), and detection failureprobability (DF). Herein, the timing accuracy and the false alarm areregarded as one criterion. The frame synchronization is affected by thelength of an auto-correlation window. However, a detection range iscontrolled to correspond to the length of the window when referring tothe criteria for the frame synchronization.

FIG. 6 depicts a process of the frame synchronization estimation whenthe CP is inserted to the short preamble instead of inserting the ISsymbol thereto in the 60 GHz WLAN system according to the exemplaryembodiment of present invention.

As shown therein, when n number of S_(m)(S₁₅, S₁₆) symbols are used forthe fine symbol timing estimation, a performance range of thecross-correlation is set to be within 16 samples. Accordingly, thedetection range of the frame synchronization is set within ±8 samplesfrom a start point of the corresponding preamble. When the length of theauto-correlation window is set to be greater than 16 samples, start ofthe frame may be found within the preamble but the detection range maynot satisfactory, thereby causing an increase of an FA. In other words,timing accuracy may cause performance degradation.

FIG. 7 to FIG. 10 respectively show length-specified auto-correlationwhen performing the frame synchronization in the 60 GHz WLAN systemaccording to the present invention.

Referring to FIG. 7 to FIG. 10, the frame may not be synchronizedbecause a peak difference between the first peak and the second peak isless than −6 dB when the length of the window is less than 16 samplesand a signal-to-noise ratio is set to be zero.

Thus, a first method is proposed to solve the foregoing problem by usingan auto-correlation window of sufficient length to extend the detectionrange.

In the first method using the auto-correlation window of a sufficientlength, a received signal is delayed by an auto-correlation delayN_(Delay), the received signal is multiplied by a conjugate complex ofthe delayed signal, a result of the multiplication is stored in a shiftregister having a window length of N_(ws), and an average value ofresults stored in the shift register is calculated to thus detect athreshold value, find a maximum position, and detect the symbol timing.

FIG. 11 illustrates a process of symbol timing detection using theauto-correlation according to the present invention, and FIG. 11 andEquation 3 summarize a process of the method.

A multiplication result 83 of a received signal y_(n) and a signaly_(n−N) _(Delay) 82 becomes an input of a moving average block 84.Herein, the signal y_(n−N) _(Delay) is a signal that is delayed by anN_(Delay) sample 81 and converted into a conjugate complex 82. A squarevalue 86 of the received signal that is delayed by the N_(Delay) sample85 is also input to a moving average block 87 to calculate an averagevalue 89 to thereby obtain a final value {circumflex over (τ)}_(n). Thefinal value {circumflex over (τ)}_(n) corresponding to a correlationcoefficient is obtained by Equation 3.

$\begin{matrix}{{\hat{\tau}}_{n} = \frac{{\sum\limits_{i = 0}^{N_{WS} - 1}{y_{k - i}y_{k - N_{Delay} - i}^{*}}}}{\sum\limits_{i = 0}^{N_{WS} - 1}{y_{k - N_{Delay} - i}^{*}}^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

Herein, it is determined that a desired signal is received when thecorrelation coefficient {circumflex over (τ)}_(n) is greater than thethreshold value.

FIG. 12 illustrates a symbol timing detection including an additionalauto-correlation designed to reduce the probability of an FA accordingto the present invention.

As shown therein, a confirmation process is repeated once to reduce theprobability of an FA. Herein, a final value {circumflex over(τ)}_(n-confirm) corresponding to the correlation coefficient forconfirmation is obtained by Equation 4.

$\begin{matrix}{{\hat{\tau}}_{n\text{-}{confirm}} = \frac{{\sum\limits_{i = 0}^{N_{WS} - 1}{y_{k - i}y_{k - N_{Delay} - N_{confirm} - i}^{*}}}}{\sum\limits_{i = 0}^{N_{WS} - 1}{y_{k - N_{Delay} - N_{confirm} - i}^{*}}^{2}}} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

FIG. 13 illustrates a process of symbol timing detection including theauto-correlation for the confirmation process according to the presentinvention. The process of the symbol timing detection of FIG. 13 issimilar to that of FIG. 11, except blocks 101 and 106 for confirmationof the auto-correlation, and a detailed description thereof will thus beomitted.

As described, the first method proposes to use a window of sufficientlength to reduce the probability of DF and FA, which however increasescomplexity of realization on the first method. Therefore, a process oftiming detection is added to the frame synchronization process toincrease timing accuracy in a second method of the present invention.For example, an auto-correlator having a window length of N samples isused to find the start point of the frame, find the highest peak of theauto-correlation within the N samples and accordingly increase thetiming accuracy within ±8 samples.

FIG. 14 is a preamble structure designed for the first method. Itillustrates a process of frame synchronization when inserting the ISsymbol instead of inserting the CP symbol in the short preamble of the60 GHz WLAN system according to the present invention. When the ISsymbol is inserted at the last period having the CP length in the timedomain and the fine symbol timing estimation employing thecross-correlation is performed thereon by using S₁₆ and the IS symbol,the corresponding performance overcomes the restriction of the detectionrange, and the detection range and the length of the auto-correlationwindow can be set without restriction.

However, a calculation amount of a cross-correlator must be consideredsince a computation period (2×|N|) of the cross-correlation is increasedas a detection range (±N) is increased.

FIG. 15 illustrates that the symbol timing estimation process is addedto the frame synchronization process in order to improve timingaccuracy, the frame synchronization having periodically repeated signalsfor radio data communication.

To find the highest peak in the auto-correlation, the auto-correlationmust produce a sharp peak by way of inserting the IS symbol in anappropriate location in the short preamble. For example, the 6^(th) Ssymbol in the current short preamble of FIG. 15 must be replaced withthe IS symbol in consideration of the window length of 64 samples and aninterval between each of the windows of 16 samples.

However, when window lengths for the auto-correlation are set to be thesame, the above-described method may cause performance degradation sincethe probability of DF is increased because of reduction of the peak inthe preamble period compared to the peak of the first method, and theprobability of FA is also increased due to an offset of the timingaccuracy. However, the method advantageously reduces the calculationamount in the fine symbol timing estimation.

The first method is simulated in the Additive White Gaussian Noise(AWGN), LOS (Line of Sight), and NLOS (Non-Line of Sight) channelmodels, and a signal-to-noise ratio (SNR) is set to be 5 dB and 10 dB,the size of a frame is set to be 200 symbols, and no clipping has beenmade.

FIG. 16 illustrates a simulation result for determining the windowlength in consideration of a calculation amount in the framesynchronization and the fine symbol timing estimation according to theexemplary embodiment of the present invention. Herein, the detectionrange is given to be ±|window size/2|, and the simulation is performedin the NLOS channel model.

FIG. 17 to FIG. 19 illustrate a result of channel-specified framesynchronization performance when the window length is set to be 64samples and the detection range is set to be 64 (±32) samples accordingto the present invention. Herein, ‘FAR’ denotes a confirmation processis included to the channel-specified frame synchronization performanceto reduce the probability of FA, and the second method is analyzed as aframe synchronization algorithm for enhancing timing accuracy.

FIG. 20 to FIG. 22 illustrate a result of the frame synchronizationalgorithm when the auto-correlation window length is set to be 64samples, the detection ranges are respectively set to be 16, 32, and 62samples in the second method.

FIG. 23 shows a table comparing the frame synchronization using thealgorithm according to the first method and the frame synchronizationwithin 16 samples as the detection range according to the second methodwhen the SNR is set to be 5 dB in both cases.

As shown in FIG. 23, a result of the comparison shows that theprobability of DF is the same in both cases, and the probability of FAis greater in the second method. However, a calculation amount in thefine symbol detection of the second method is four times less than thatof the first method.

FIG. 24 shows a table comparing the frame synchronization using thealgorithm according to the first method and the frame synchronizationwithin 64 samples as the detection range according to the second methodwhen the SNR is set to be 5 dB in both cases. In this comparison, theprobability of FA is lower in the second method, but it is preferred toconsider a method for expanding the detection range of the first method.

According to the present invention, the start point of each frame isdetected when a data frame/packet is transmitted/received in the 60 GHzWLAN system to thereby obtain high credibility when a receiverdemodulates a signal, and increase capacity of the 60 GHz WLAN system.

In addition, according to the present invention, implementation of the60 GHz WLAN system is less complicated and the cost for manufacturing anintegral circuit IC is reduced. Further, when the present invention isapplied to a wireless terminal which is sensitive to power consumption,the wireless terminal can be used without recharging it for acomparatively longer time.

While this invention has been described in connection with what ispresently considered to be the most practical and preferred embodiment,it is to be understood that the invention is not limited to thedisclosed embodiment, but, on the contrary, is intended to cover variousmodifications and equivalent arrangements included within the spirit andscope of the appended claims.

What is claimed is:
 1. A method performed by a receiver of a wirelesslocal area network system, the method comprising: receiving a preambleof a frame; and using the preamble for synchronization and channelestimation, wherein: the preamble includes a short preamble used for thesynchronization and a channel estimation preamble that is used for thechannel estimation and is arranged after the short preamble; the shortpreamble includes S symbols in which an S symbol is repeated 16 time andthat is arranged at the starting point and an IS symbol arranged afterthe S symbols; and the IS symbol has the same length as the S symbol andis 180°-phased with respect to the S symbol.
 2. The method of claim 1,wherein the channel estimation preamble comprises a first symbol, asecond symbol and a third symbol that are arranged after the shortpreamble.
 3. The method of claim 2, wherein a length of the first symbolis equal to a length of the second symbol, and a length of the thirdsymbol is shorter than half of the length of the first symbol.
 4. Anapparatus of a wireless local area network system, the methodcomprising: one or more processors that uses a preamble of a frame forsynchronization and channel estimation, wherein: the preamble includes ashort preamble used for the synchronization and a channel estimationpreamble that is used for the channel estimation and is arranged afterthe short preamble; the short preamble includes S symbols in which an Ssymbol is repeated 16 time and that is arranged at the starting pointand an IS symbol arranged after the S symbols; and the IS symbol has thesame length as the S symbol and is 180°-phased with respect to the Ssymbol.
 5. The apparatus of claim 4, wherein the channel estimationpreamble comprises a first symbol, a second symbol and a third symbolthat are arranged after the short preamble.
 6. The apparatus of claim 5,wherein a length of the first symbol is equal to a length of the secondsymbol, and a length of the third symbol is shorter than half of thelength of the first symbol.
 7. A non-transitory recording medium thatstores instructions to a receiver of a wireless local area networksystem to execute: receiving a preamble of a frame; and using thepreamble for synchronization and channel estimation, wherein: thepreamble includes a short preamble used for the synchronization and achannel estimation preamble that is used for the channel estimation andis arranged after the short preamble; the short preamble includes Ssymbols in which an S symbol is repeated 16 time and that is arranged atthe starting point and an IS symbol arranged after the S symbols; andthe IS symbol has the same length as the S symbol and is 180°-phasedwith respect to the S symbol.
 8. The non-transitory recording medium ofclaim 7, wherein the channel estimation preamble comprises a firstsymbol, a second symbol and a third symbol that are arranged after theshort preamble.
 9. The non-transitory recording medium of claim 8,wherein a length of the first symbol is equal to a length of the secondsymbol, and a length of the third symbol is shorter than half of thelength of the first symbol.